Lattices in the cohomology of Shimura curves

We prove the main conjectures of Breuil (J Reine Angew Math, 2012 ) (including a generalisation from the principal series to the cuspidal case) and Dembélé (J Reine Angew Math, 2012 ), subject to a mild global hypothesis that we make in order to apply certain R = T theorems. More precisely, we prove...

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Bibliographic Details
Published in:Inventiones mathematicae 2015-04, Vol.200 (1), p.1-96
Main Authors: Emerton, Matthew, Gee, Toby, Savitt, David
Format: Article
Language:English
Subjects:
Deformation
Lattices
Mathematical analysis
Mathematics
Mathematics and Statistics
Patching
Philosophy
Representations
Texts
Theorems
Two dimensional
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Summary:We prove the main conjectures of Breuil (J Reine Angew Math, 2012 ) (including a generalisation from the principal series to the cuspidal case) and Dembélé (J Reine Angew Math, 2012 ), subject to a mild global hypothesis that we make in order to apply certain R = T theorems. More precisely, we prove a multiplicity one result for the mod p cohomology of a Shimura curve at Iwahori level, and we show that certain apparently globally defined lattices in the cohomology of Shimura curves are determined by the corresponding local p -adic Galois representations. We also indicate a new proof of the Buzzard–Diamond–Jarvis conjecture in generic cases. Our main tools are the geometric Breuil–Mézard philosophy developed in Emerton and Gee (J Inst Math Jussieu, 2012 ), and a new and more functorial perspective on the Taylor–Wiles–Kisin patching method. Along the way, we determine the tamely potentially Barsotti–Tate deformation rings of generic two-dimensional mod p representations, generalising a result of Breuil and Mézard (Bull Soc Math de France, 2012 ) in the principal series case.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-014-0517-0